Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions

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November 2013, 12(6): 2697-2713. doi: 10.3934/cpaa.2013.12.2697

Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions

Mateus Balbino Guimarães ^1, and Rodrigo da Silva Rodrigues ^1,

1.	Universidade Federal de São Carlos, Departamento de Matemática, Rod. Washington Luís, Km 235, CEP. 13565-905, São Carlos, SP, Brazil, Brazil

Received September 2012 Revised October 2012 Published May 2013

Abstract
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In this article we establish the existence and nonexistence of a weak solution to singular elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions.

Keywords: Variational methods, existence of solution, critical Caffarelli-Kohn-Nirenberg exponent, nonexistence of solution, singular and sign-changing weights..

Mathematics Subject Classification: Primary: 35A15, 35B33, 35B25; Secondary: 35J6.

Citation: Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697

References:

[1]	M. Bouchekif and A. Matallah, Singular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions,, Electronic Journal of Differential Equations, 2010 (2010), 1.
[2]	H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc.Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999.
[3]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405.
[4]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights,, Compos. Math., 53 (1984), 259.
[5]	N. Chaudhuri and M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients,, Royal Society of Edinburgh, 131A (2001), 1275. doi: 10.1017/S0308210500001396.
[6]	K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality,, J. London Math. Soc., 2 (1993), 137. doi: 10.1112/jlms/s2-48.1.137.
[7]	L. C. Evans, "Partial Differential Equations,'', Graduate studies in mathematics 19, (1998).
[8]	A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations,, J. Differential Equations, 177 (2001), 494. doi: 10.1006/jdeq.2000.3999.
[9]	P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms,, Nonlinear Anal., 61 (2005), 735. doi: 10.1016/j.na.2005.01.030.
[10]	X. J. Huang, X. P. Wu and C. L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents,, Nonlinear Anal., 74 (2011), 2602. doi: 10.1016/j.na.2010.12.015.
[11]	E. Jannelli, The role played by space dimension in elliptic critical problems,, J. Differential Equations, 156 (1999), 407. doi: 10.1006/jdeq.1998.3589.
[12]	M. Lin, Some further results for a class of weighted nonlinear elliptic equations,, J. Math. Anal. Appl., 337 (2008), 537. doi: 10.1016/j.jmaa.2007.04.034.
[13]	O. H. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: 10.1016/S0362-546X(96)00087-9.
[14]	R. S. Rodrigues, On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function,, Nonlinear Anal., 73 (2010), 857. doi: 10.1016/j.na.2010.03.053.
[15]	M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer-Verlag, (1990).
[16]	B. J. Xuan, S. Su and Y. Yan, Existence results for Brézis-Nirenberg problems with Hardy potential and singular coefficients,, Nonlinear Anal., 67 (2007), 2091. doi: 10.1016/j.na.2006.09.018.
[17]	B. J. Xuan, The solvability of quasilinear Brézis-Nirenberg-type problems with singular weights,, Nonlinear Anal., 62 (2005), 703. doi: 10.1016/j.na.2005.03.095.

show all references

References:

[1]	M. Bouchekif and A. Matallah, Singular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions,, Electronic Journal of Differential Equations, 2010 (2010), 1.
[2]	H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc.Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999.
[3]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405.
[4]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights,, Compos. Math., 53 (1984), 259.
[5]	N. Chaudhuri and M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients,, Royal Society of Edinburgh, 131A (2001), 1275. doi: 10.1017/S0308210500001396.
[6]	K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality,, J. London Math. Soc., 2 (1993), 137. doi: 10.1112/jlms/s2-48.1.137.
[7]	L. C. Evans, "Partial Differential Equations,'', Graduate studies in mathematics 19, (1998).
[8]	A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations,, J. Differential Equations, 177 (2001), 494. doi: 10.1006/jdeq.2000.3999.
[9]	P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms,, Nonlinear Anal., 61 (2005), 735. doi: 10.1016/j.na.2005.01.030.
[10]	X. J. Huang, X. P. Wu and C. L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents,, Nonlinear Anal., 74 (2011), 2602. doi: 10.1016/j.na.2010.12.015.
[11]	E. Jannelli, The role played by space dimension in elliptic critical problems,, J. Differential Equations, 156 (1999), 407. doi: 10.1006/jdeq.1998.3589.
[12]	M. Lin, Some further results for a class of weighted nonlinear elliptic equations,, J. Math. Anal. Appl., 337 (2008), 537. doi: 10.1016/j.jmaa.2007.04.034.
[13]	O. H. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: 10.1016/S0362-546X(96)00087-9.
[14]	R. S. Rodrigues, On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function,, Nonlinear Anal., 73 (2010), 857. doi: 10.1016/j.na.2010.03.053.
[15]	M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer-Verlag, (1990).
[16]	B. J. Xuan, S. Su and Y. Yan, Existence results for Brézis-Nirenberg problems with Hardy potential and singular coefficients,, Nonlinear Anal., 67 (2007), 2091. doi: 10.1016/j.na.2006.09.018.
[17]	B. J. Xuan, The solvability of quasilinear Brézis-Nirenberg-type problems with singular weights,, Nonlinear Anal., 62 (2005), 703. doi: 10.1016/j.na.2005.03.095.

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